A nonlinear parametric model reduction method for efficient gear contact simulations

Abstract

A novel nonlinear parametric model order reduction technique for the solution of contact problems in flexible multibody dynamics is presented. These problems are characterized by significant variations in the location and size of the contact area and typically require high-dimensional finite element models having multiple inputs and outputs to be solved. The presented technique draws from the fields of nonlinear and parametric model reduction to construct a reduced-order model whose dimensions are insensitive to the dimensions of the full-order model. The solution of interest is approximated in a lower-dimensional subspace spanned by a constant set of eigenvectors augmented with a parameter-dependent set of global contact shapes. The latter represent deformation patterns of the interacting bodies obtained from a series of static contact analyses. The set of global contact shapes is parameterized with respect to the system configuration and therefore continuously varies in time. An energy-consistent formulation is assured by explicitly taking into account the dynamic parameter variability in the derivation of the equations of motion. The performance of the novel technique is demonstrated by simulating a dynamic gear contact problem and comparing results against traditional model reduction techniques as well as commercial nonlinear finite element software. Copyright NLPMOR METHOD FOR EFFICIENT GEAR SIMULATIONS 1163 meshes. These approaches separate the bulk deformation of the interacting components from the nonlinear local displacement field at the contact surfaces, computing the former using linear FE on a coarse mesh and obtaining the latter from classical contact theory. Most notably, Vijayakar [3] combines traditional FE with the Boussinesq solution for a point-load acting on an infinite halfspace, whereas Andersson and Vedmar [2] compute the local deformation due to Hertzian contact pressure using a formula derived by Weber and Banaschek [4]. Although these approaches require significantly less degrees of freedom as compared to traditional FE-based contact simulations, the procedure of matching the analytical and FE solutions at a certain depth below the contact surface is computationally involved, resulting in relatively high costs per time increment [5]. Moreover, an energy-consistent adaptation of these approaches to the formalism of flexible multibody dynamics (FMBS) has yet to be pursued. Finally, the question as to whether an infinite half-space serves as a good approximation of the actual contact zone is highly case-specific.The use of conventional FE to represent flexible bodies in multibody simulations can be rendered practical with the aid of model order reduction (MOR) techniques. These techniques originated from the fields of control theory [6] and structural dynamics [7] and were primarily designed for reducing the sizes of, respectively, first-order and second-order linear systems. Using the floating frame of reference formulation [8], the linear elastic deformation of a flexible body is separate...

12 Model order reduction and digital twins

“…Drivetrains contain a multitude of components, including bearings, gears, clutches, and spline connections that are known to behave nonlinearly and contain multiple contacts between flexible objects. While several MOR methods have been applied largely and successfully [32] in the field of flexible multibody simulations [97] in both academic and industrial settings with a large growing body of literature, MOR methods dedicated to the field of contact mechanics have been only recently explored [12,103]. Moreover the developed methods often target high-dynamic contact mechanics simulations with fully flexible bodies and dynamic interactions with flexible eigenmodes of the structure [12].…”

“…In recent years several methods have been presented to maintain a high level of accuracy -similar to nonlinear finite element full-order dynamic computations -but drastically limit the impact of the above-mentioned issues. In particular, the following works [11,12] obtain very good results in terms of speedup and memory usage while losing only a fraction of the accuracy obtained with nonlinear finite element problems. The field of hyperreduction [18,31] is also exploited to tackle the contact detection problem with very promising results.…”

2 Model order reduction in mechanical engineering

“…Today, the successful applications of model order reduction (MOR) in mechanical engineering deal with different system and problem classes from different physical domains, like: -structural and multibody dynamics, modeled by linear or nonlinear differential equations [18,17,21,67,69,35,34,53,3,31,91,90,5,104,19,49,47,95,110]; -fluid dynamics, including fluid-structure interaction and aerodynamics [93,26,89,88,83,2,31]; -thermo-mechanical, thermo-fluid, thermo-acoustic, and thermo-electrical systems [68,12,14,11,97,75,54,46].…”

A ‘nodeless’ dual superelement formulation for structural and multibody dynamics application to reduction of contact problems

“…Indeed, it is shown in the sequel of the paper that the contribution of static modes to properly grasp the frequency content of the model is important. In a recent contribution , the technique of global modal parameterization has been introduced to construct a variable static modal basis that does not suffer from the appearance of spurious transients due to mode switching. The efficiency of such time‐dependent parametric interpolation has been demonstrated on the complex problem of dynamic stress recovery in gear systems.…”